hyperbolic_system.h

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//
// SPDX-License-Identifier: Apache-2.0
// [LANL Copyright Statement]
// Copyright (C) 2020 - 2024 by the ryujin authors
// Copyright (C) 2023 - 2024 by Triad National Security, LLC
//
 
#pragma once
 
#include <compile_time_options.h>
 
#include <convenience_macros.h>
#include <deal.II/base/config.h>
#include <discretization.h>
#include <multicomponent_vector.h>
#include <openmp.h>
#include <patterns_conversion.h>
#include <simd.h>
#include <state_vector.h>
 
#include <deal.II/base/parameter_acceptor.h>
#include <deal.II/base/tensor.h>
 
#include <array>
 
namespace ryujin
{
  namespace ShallowWater
  {
    template <int dim, typename Number>
    class HyperbolicSystemView;
 
    class HyperbolicSystem final : public dealii::ParameterAcceptor
    {
    public:
      static inline const std::string problem_name = "Shallow water equations";
 
      HyperbolicSystem(const std::string &subsection = "/HyperbolicSystem");
 
      template <int dim, typename Number>
      auto view() const
      {
        return HyperbolicSystemView<dim, Number>{*this};
      }
 
    private:
      double gravity_;
      double manning_friction_coefficient_;
 
      double reference_water_depth_;
      double dry_state_relaxation_small_;
      double dry_state_relaxation_large_;
 
      template <int dim, typename Number>
      friend class HyperbolicSystemView;
    }; /* HyperbolicSystem */
 
 
    template <int dim, typename Number>
    class HyperbolicSystemView
    {
    public:
      HyperbolicSystemView(const HyperbolicSystem &hyperbolic_system)
          : hyperbolic_system_(hyperbolic_system)
      {
      }
 
      template <int dim2, typename Number2>
      auto view() const
      {
        return HyperbolicSystemView<dim2, Number2>{hyperbolic_system_};
      }
 
      using ScalarNumber = typename get_value_type<Number>::type;
 
 
      DEAL_II_ALWAYS_INLINE inline ScalarNumber gravity() const
      {
        return hyperbolic_system_.gravity_;
      }
 
      DEAL_II_ALWAYS_INLINE inline ScalarNumber
      manning_friction_coefficient() const
      {
        return hyperbolic_system_.manning_friction_coefficient_;
      }
 
      DEAL_II_ALWAYS_INLINE inline ScalarNumber reference_water_depth() const
      {
        return hyperbolic_system_.reference_water_depth_;
      }
 
      DEAL_II_ALWAYS_INLINE inline ScalarNumber
      dry_state_relaxation_small() const
      {
        return hyperbolic_system_.dry_state_relaxation_small_;
      }
 
      DEAL_II_ALWAYS_INLINE inline ScalarNumber
      dry_state_relaxation_large() const
      {
        return hyperbolic_system_.dry_state_relaxation_large_;
      }
 
 
 
    private:
      const HyperbolicSystem &hyperbolic_system_;
 
    public:
 
 
      static constexpr unsigned int problem_dimension = 1 + dim;
 
      using state_type = dealii::Tensor<1, problem_dimension, Number>;
 
      using flux_type =
          dealii::Tensor<1, problem_dimension, dealii::Tensor<1, dim, Number>>;
 
      using flux_contribution_type = std::tuple<state_type, Number>;
 
      static inline const auto component_names =
          []() -> std::array<std::string, problem_dimension> {
        if constexpr (dim == 1)
          return {"h", "m"};
        else if constexpr (dim == 2)
          return {"h", "m_1", "m_2"};
        else if constexpr (dim == 3)
          return {"h", "m_1", "m_2", "m_3"};
        __builtin_trap();
      }();
 
      static inline const auto primitive_component_names =
          []() -> std::array<std::string, problem_dimension> {
        if constexpr (dim == 1)
          return {"h", "v"};
        else if constexpr (dim == 2)
          return {"h", "v_1", "v_2"};
        else if constexpr (dim == 3)
          return {"h", "v_1", "v_2", "v_3"};
        __builtin_trap();
      }();
 
      static constexpr unsigned int n_precomputed_values = 2;
 
      using precomputed_type = std::array<Number, n_precomputed_values>;
 
      static inline const auto precomputed_names =
          std::array<std::string, n_precomputed_values>{"eta_m", "h_star"};
 
      static constexpr unsigned int n_initial_precomputed_values = 1;
 
      using initial_precomputed_type =
          std::array<Number, n_initial_precomputed_values>;
 
      static inline const auto initial_precomputed_names =
          std::array<std::string, n_initial_precomputed_values>{"bathymetry"};
 
      using StateVector = Vectors::
          StateVector<ScalarNumber, problem_dimension, n_precomputed_values>;
 
      using HyperbolicVector =
          Vectors::MultiComponentVector<ScalarNumber, problem_dimension>;
 
      using PrecomputedVector =
          Vectors::MultiComponentVector<ScalarNumber, n_precomputed_values>;
 
      using InitialPrecomputedVector =
          Vectors::MultiComponentVector<ScalarNumber,
                                        n_initial_precomputed_values>;
 
 
 
      static constexpr unsigned int n_precomputation_cycles = 1;
 
      template <typename DISPATCH, typename SPARSITY>
      void precomputation_loop(unsigned int cycle,
                               const DISPATCH &dispatch_check,
                               const SPARSITY &sparsity_simd,
                               StateVector &state_vector,
                               unsigned int left,
                               unsigned int right,
                               const bool skip_constrained_dofs = true) const;
 
 
 
      static Number water_depth(const state_type &U);
 
      Number inverse_water_depth_mollified(const state_type &U) const;
 
      Number water_depth_sharp(const state_type &U) const;
 
      Number inverse_water_depth_sharp(const state_type &U) const;
 
      Number filter_dry_water_depth(const Number &h) const;
 
      static dealii::Tensor<1, dim, Number> momentum(const state_type &U);
 
      Number kinetic_energy(const state_type &U) const;
 
      Number pressure(const state_type &U) const;
 
      Number speed_of_sound(const state_type &U) const;
 
      Number mathematical_entropy(const state_type &U) const;
 
      state_type mathematical_entropy_derivative(const state_type &U) const;
 
      bool is_admissible(const state_type &U) const;
 
 
 
      template <int component>
      state_type prescribe_riemann_characteristic(
          const state_type &U,
          const state_type &U_bar,
          const dealii::Tensor<1, dim, Number> &normal) const;
 
      template <typename Lambda>
      state_type
      apply_boundary_conditions(const dealii::types::boundary_id id,
                                const state_type &U,
                                const dealii::Tensor<1, dim, Number> &normal,
                                const Lambda &get_dirichlet_data) const;
 
 
 
      flux_type f(const state_type &U) const;
 
      flux_type g(const state_type &U) const;
 
      state_type star_state(const state_type &U,
                            const Number &Z_left,
                            const Number &Z_right) const;
 
      std::array<state_type, 2>
      equilibrated_states(const flux_contribution_type &,
                          const flux_contribution_type &) const;
 
      flux_contribution_type
      flux_contribution(const PrecomputedVector &pv,
                        const InitialPrecomputedVector &piv,
                        const unsigned int i,
                        const state_type &U_i) const;
 
      flux_contribution_type
      flux_contribution(const PrecomputedVector &pv,
                        const InitialPrecomputedVector &piv,
                        const unsigned int *js,
                        const state_type &U_j) const;
 
      state_type
      flux_divergence(const flux_contribution_type &flux_i,
                      const flux_contribution_type &flux_j,
                      const dealii::Tensor<1, dim, Number> &c_ij) const;
 
      static constexpr bool have_high_order_flux = true;
 
      state_type high_order_flux_divergence(
          const flux_contribution_type &flux_i,
          const flux_contribution_type &flux_j,
          const dealii::Tensor<1, dim, Number> &c_ij) const;
 
      state_type affine_shift(const flux_contribution_type &flux_i,
                              const flux_contribution_type &flux_j,
                              const dealii::Tensor<1, dim, Number> &c_ij,
                              const Number &d_ij) const;
 
 
 
      static constexpr bool have_source_terms = true;
 
      state_type manning_friction(const state_type &U,
                                  const Number &h_star,
                                  const ScalarNumber tau) const;
 
      state_type nodal_source(const PrecomputedVector &pv,
                              const unsigned int i,
                              const state_type &U_i,
                              const ScalarNumber tau) const;
 
      state_type nodal_source(const PrecomputedVector &pv,
                              const unsigned int *js,
                              const state_type &U_j,
                              const ScalarNumber tau) const;
 
 
 
      template <typename ST>
      state_type expand_state(const ST &state) const;
 
      template <typename ST>
      state_type from_initial_state(const ST &initial_state) const;
 
      state_type from_primitive_state(const state_type &primitive_state) const;
 
      state_type to_primitive_state(const state_type &state) const;
 
      template <typename Lambda>
      state_type apply_galilei_transform(const state_type &state,
                                         const Lambda &lambda) const;
 
    }; /* HyperbolicSystemView */
 
 
    /*
     * -------------------------------------------------------------------------
     * Inline definitions
     * -------------------------------------------------------------------------
     */
 
 
    inline HyperbolicSystem::HyperbolicSystem(const std::string &subsection)
        : ParameterAcceptor(subsection)
    {
      gravity_ = 9.81;
      add_parameter("gravity", gravity_, "Gravitational constant [m/s^2]");
 
      manning_friction_coefficient_ = 0.;
      add_parameter("manning friction coefficient",
                    manning_friction_coefficient_,
                    "Roughness coefficient for friction source");
 
      reference_water_depth_ = 1.;
      add_parameter("reference water depth",
                    reference_water_depth_,
                    "Problem specific water depth reference");
 
      dry_state_relaxation_small_ = 1.e2;
      add_parameter("dry state relaxation small",
                    dry_state_relaxation_small_,
                    "Problem specific dry-state relaxation parameter");
 
      dry_state_relaxation_large_ = 1.e4;
      add_parameter("dry state relaxation large",
                    dry_state_relaxation_large_,
                    "Problem specific dry-state relaxation parameter");
    }
 
 
    template <int dim, typename Number>
    template <typename DISPATCH, typename SPARSITY>
    DEAL_II_ALWAYS_INLINE inline void
    HyperbolicSystemView<dim, Number>::precomputation_loop(
        unsigned int cycle [[maybe_unused]],
        const DISPATCH &dispatch_check,
        const SPARSITY &sparsity_simd,
        StateVector &state_vector,
        unsigned int left,
        unsigned int right,
        const bool skip_constrained_dofs /*= true*/) const
    {
      Assert(cycle == 0, dealii::ExcInternalError());
 
      const auto &U = std::get<0>(state_vector);
      auto &precomputed = std::get<1>(state_vector);
 
      /* We are inside a thread parallel context */
 
      unsigned int stride_size = get_stride_size<Number>;
 
      RYUJIN_OMP_FOR
      for (unsigned int i = left; i < right; i += stride_size) {
 
        /* Skip constrained degrees of freedom: */
        const unsigned int row_length = sparsity_simd.row_length(i);
        if (skip_constrained_dofs && row_length == 1)
          continue;
 
        dispatch_check(i);
 
        const auto U_i = U.template get_tensor<Number>(i);
 
        const auto eta_m = mathematical_entropy(U_i);
 
        const auto h_sharp = water_depth_sharp(U_i);
        const auto h_star = ryujin::pow(h_sharp, ScalarNumber(4. / 3.));
 
        const precomputed_type prec_i{eta_m, h_star};
        precomputed.template write_tensor<Number>(prec_i, i);
      }
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::water_depth(const state_type &U)
    {
      return U[0];
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::inverse_water_depth_mollified(
        const state_type &U) const
    {
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
 
      const Number h_cutoff_mollified =
          reference_water_depth() * dry_state_relaxation_large() * Number(eps);
 
      const Number h = water_depth(U);
      const Number h_pos = positive_part(water_depth(U));
      const Number h_max = std::max(h, h_cutoff_mollified);
      const Number denom = h * h + h_max * h_max;
      return ScalarNumber(2.) * h_pos / denom;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::water_depth_sharp(
        const state_type &U) const
    {
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
 
      const Number h_cutoff_small =
          reference_water_depth() * dry_state_relaxation_small() * Number(eps);
 
      const Number h = water_depth(U);
      const Number h_max = std::max(h, h_cutoff_small);
      return h_max;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::inverse_water_depth_sharp(
        const state_type &U) const
    {
      return ScalarNumber(1.) / water_depth_sharp(U);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::filter_dry_water_depth(
        const Number &h) const
    {
      using ScalarNumber = typename get_value_type<Number>::type;
      constexpr ScalarNumber eps = std::numeric_limits<ScalarNumber>::epsilon();
 
      const Number h_cutoff_large =
          reference_water_depth() * dry_state_relaxation_large() * Number(eps);
 
      return dealii::compare_and_apply_mask<dealii::SIMDComparison::less_than>(
          std::abs(h), h_cutoff_large, Number(0.), h);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline dealii::Tensor<1, dim, Number>
    HyperbolicSystemView<dim, Number>::momentum(const state_type &U)
    {
      dealii::Tensor<1, dim, Number> result;
 
      for (unsigned int i = 0; i < dim; ++i)
        result[i] = U[1 + i];
      return result;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::kinetic_energy(const state_type &U) const
    {
      const auto h = water_depth(U);
      const auto vel = momentum(U) * inverse_water_depth_sharp(U);
 
      /* KE = 1/2 h |v|^2 */
      return ScalarNumber(0.5) * h * vel.norm_square();
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::pressure(const state_type &U) const
    {
      const Number h_sqd = U[0] * U[0];
 
      /* p = 1/2 g h^2 */
      return ScalarNumber(0.5) * gravity() * h_sqd;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::speed_of_sound(const state_type &U) const
    {
      /* c^2 = g * h */
      return std::sqrt(gravity() * U[0]);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline Number
    HyperbolicSystemView<dim, Number>::mathematical_entropy(
        const state_type &U) const
    {
      const auto p = pressure(U);
      const auto k_e = kinetic_energy(U);
      return p + k_e;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::mathematical_entropy_derivative(
        const state_type &U) const -> state_type
    {
      /*
       * With
       *   eta = 1/2 g h^2 + 1/2 |m|^2 / h
       *
       * we get
       *
       *   eta' = (g h - 1/2 |vel|^2, vel)
       *
       * where vel = m / h
       */
 
      state_type result;
 
      const Number &h = U[0];
      const auto vel = momentum(U) * inverse_water_depth_sharp(U);
 
      // water depth component
      result[0] = gravity() * h - ScalarNumber(0.5) * vel.norm_square();
 
      // momentum components
      for (unsigned int i = 0; i < dim; ++i) {
        result[1 + i] = vel[i];
      }
 
      return result;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline bool
    HyperbolicSystemView<dim, Number>::is_admissible(const state_type &U) const
    {
      const auto h = filter_dry_water_depth(water_depth(U));
 
      constexpr auto gte = dealii::SIMDComparison::greater_than_or_equal;
      const auto test = dealii::compare_and_apply_mask<gte>(
          h, Number(0.), Number(0.), Number(-1.));
 
#ifdef DEBUG_OUTPUT
      if (!(test == Number(0.))) {
        std::cout << std::fixed << std::setprecision(16);
        std::cout << "Bounds violation: Negative state [h] detected!\n";
        std::cout << "\t\th: " << h << "\n" << std::endl;
        __builtin_trap();
      }
#endif
 
      return (test == Number(0.));
    }
 
 
    template <int dim, typename Number>
    template <int component>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::prescribe_riemann_characteristic(
        const state_type &U,
        const state_type &U_bar,
        const dealii::Tensor<1, dim, Number> &normal) const -> state_type
    {
      /* Note that U_bar are the dirichlet values that are prescribed */
      static_assert(component == 1 || component == 2,
                    "component has to be 1 or 2");
 
      using ScalarNumber = typename get_value_type<Number>::type;
 
      const auto m = momentum(U);
      const auto a = speed_of_sound(U);
      const auto vn = m * normal * inverse_water_depth_sharp(U);
 
      const auto m_bar = momentum(U_bar);
      const auto a_bar = speed_of_sound(U_bar);
      const auto vn_bar = m_bar * normal * inverse_water_depth_sharp(U_bar);
 
      /* First Riemann characteristic: v * n - 2 * a */
 
      const auto R_1 = component == 1 ? vn_bar - ScalarNumber(2.) * a_bar
                                      : vn - ScalarNumber(2.) * a;
 
      /* Second Riemann characteristic: v * n + 2 * a */
 
      const auto R_2 = component == 2 ? vn_bar + ScalarNumber(2.) * a_bar
                                      : vn + ScalarNumber(2.) * a;
 
      const auto vperp = m * inverse_water_depth_sharp(U) - vn * normal;
 
      const auto vn_new = ScalarNumber(0.5) * (R_1 + R_2);
 
      const auto h_new =
          ryujin::fixed_power<2>((R_2 - R_1) / ScalarNumber(4.)) / gravity();
 
      state_type U_new;
      U_new[0] = h_new;
      for (unsigned int d = 0; d < dim; ++d) {
        U_new[1 + d] = h_new * (vn_new * normal + vperp)[d];
      }
 
      return U_new;
    }
 
 
    template <int dim, typename Number>
    template <typename Lambda>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::apply_boundary_conditions(
        const dealii::types::boundary_id id,
        const state_type &U,
        const dealii::Tensor<1, dim, Number> &normal,
        const Lambda &get_dirichlet_data) const -> state_type
    {
      state_type result = U;
 
      if (id == Boundary::dirichlet) {
        result = get_dirichlet_data();
 
      } else if (id == Boundary::dirichlet_momentum) {
        /* Only enforce Dirichlet conditions on the momentum: */
        auto m_dirichlet = momentum(get_dirichlet_data());
        for (unsigned int k = 0; k < dim; ++k)
          result[k + 1] = m_dirichlet[k];
 
      } else if (id == Boundary::slip) {
        auto m = momentum(U);
        m -= 1. * (m * normal) * normal;
        for (unsigned int k = 0; k < dim; ++k)
          result[k + 1] = m[k];
 
      } else if (id == Boundary::no_slip) {
        for (unsigned int k = 0; k < dim; ++k)
          result[k + 1] = Number(0.);
 
      } else if (id == Boundary::dynamic) {
        /*
         * On dynamic boundary conditions, we distinguish four cases:
         *
         *  - supersonic inflow: prescribe full state
         *  - subsonic inflow:
         *      decompose into Riemann invariants and leave R_2
         *      characteristic untouched.
         *  - supersonic outflow: do nothing
         *  - subsonic outflow:
         *      decompose into Riemann invariants and prescribe incoming
         *      R_1 characteristic.
         */
        const auto m = momentum(U);
        const auto h_inverse = inverse_water_depth_sharp(U);
        const auto a = speed_of_sound(U);
        const auto vn = m * normal * h_inverse;
 
        /* Supersonic inflow: */
        if (vn < -a) {
          result = get_dirichlet_data();
        }
 
        /* Subsonic inflow: */
        if (vn >= -a && vn <= 0.) {
          const auto U_dirichlet = get_dirichlet_data();
          result = prescribe_riemann_characteristic<2>(U_dirichlet, U, normal);
        }
 
        /* Subsonic outflow: */
        if (vn > 0. && vn <= a) {
          const auto U_dirichlet = get_dirichlet_data();
          result = prescribe_riemann_characteristic<1>(U, U_dirichlet, normal);
        }
 
        /* Supersonic outflow: do nothing, i.e., keep U as is */
 
      } else {
        AssertThrow(false, dealii::ExcNotImplemented());
      }
 
      return result;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::f(const state_type &U) const -> flux_type
    {
      const auto h_inverse = inverse_water_depth_sharp(U);
      const auto m = momentum(U);
      const auto p = pressure(U);
 
      flux_type result;
 
      result[0] = (m * h_inverse) * U[0];
      for (unsigned int i = 0; i < dim; ++i) {
        result[1 + i] = (m * h_inverse) * m[i];
        result[1 + i][i] += p;
      }
      return result;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::g(const state_type &U) const -> flux_type
    {
      const auto h_inverse = inverse_water_depth_sharp(U);
      const auto m = momentum(U);
 
      flux_type result;
 
      result[0] = (m * h_inverse) * U[0];
      for (unsigned int i = 0; i < dim; ++i) {
        result[1 + i] = (m * h_inverse) * m[i];
      }
      return result;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::star_state(const state_type &U,
                                                  const Number &Z_left,
                                                  const Number &Z_right) const
        -> state_type
    {
      const Number Z_max = std::max(Z_left, Z_right);
      const Number h = water_depth(U);
      const Number H_star = std::max(Number(0.), h + Z_left - Z_max);
 
      return U * H_star * inverse_water_depth_mollified(U);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::equilibrated_states(
        const flux_contribution_type &flux_i,
        const flux_contribution_type &flux_j) const -> std::array<state_type, 2>
    {
      const auto &[U_i, Z_i] = flux_i;
      const auto &[U_j, Z_j] = flux_j;
 
      const auto U_star_ij = star_state(U_i, Z_i, Z_j);
      const auto U_star_ji = star_state(U_j, Z_j, Z_i);
 
      return {U_star_ij, U_star_ji};
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::flux_contribution(
        const PrecomputedVector & /*pv*/,
        const InitialPrecomputedVector &piv,
        const unsigned int i,
        const state_type &U_i) const -> flux_contribution_type
    {
      const auto Z_i = piv.template get_tensor<Number>(i)[0];
      return {U_i, Z_i};
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::flux_contribution(
        const PrecomputedVector & /*pv*/,
        const InitialPrecomputedVector &piv,
        const unsigned int *js,
        const state_type &U_j) const -> flux_contribution_type
    {
      const auto Z_j = piv.template get_tensor<Number>(js)[0];
      return {U_j, Z_j};
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::flux_divergence(
        const flux_contribution_type &flux_i,
        const flux_contribution_type &flux_j,
        const dealii::Tensor<1, dim, Number> &c_ij) const -> state_type
    {
      const auto &[U_i, Z_i] = flux_i;
      const auto &[U_star_ij, U_star_ji] = equilibrated_states(flux_i, flux_j);
 
      const auto H_i = water_depth(U_i);
      const auto H_star_ij = water_depth(U_star_ij);
      const auto H_star_ji = water_depth(U_star_ji);
 
      const auto g_i = g(U_star_ij);
      const auto g_j = g(U_star_ji);
 
      auto result = -add(g_i, g_j);
 
      const auto factor =
          (ScalarNumber(0.5) * (H_star_ji * H_star_ji - H_star_ij * H_star_ij) +
           H_i * H_i) *
          gravity();
 
      for (unsigned int i = 0; i < dim; ++i) {
        result[1 + i][i] -= factor;
      }
 
      return contract(result, c_ij);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::high_order_flux_divergence(
        const flux_contribution_type &flux_i,
        const flux_contribution_type &flux_j,
        const dealii::Tensor<1, dim, Number> &c_ij) const -> state_type
    {
      const auto &[U_i, Z_i] = flux_i;
      const auto &[U_j, Z_j] = flux_j;
 
      const auto H_i = water_depth(U_i);
      const auto H_j = water_depth(U_j);
 
      const auto g_i = g(U_i);
      const auto g_j = g(U_j);
 
      auto result = -add(g_i, g_j);
 
      const auto factor = gravity() * H_i * (H_j + Z_j - Z_i);
      for (unsigned int i = 0; i < dim; ++i) {
        result[1 + i][i] -= factor;
      }
 
      return contract(result, c_ij);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::affine_shift(
        const flux_contribution_type &flux_i,
        const flux_contribution_type &flux_j,
        const dealii::Tensor<1, dim, Number> &c_ij,
        const Number &d_ij) const -> state_type
    {
      const auto &[U_i, Z_i] = flux_i;
      const auto &[U_j, Z_j] = flux_j;
      const auto U_star_ij = star_state(U_i, Z_i, Z_j);
 
      const auto h_inverse = inverse_water_depth_sharp(U_i);
      const auto m = momentum(U_i);
      const auto factor = ScalarNumber(2.) * (d_ij + h_inverse * (m * c_ij));
 
      return -factor * (U_star_ij - U_i);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::manning_friction(
        const state_type &U, const Number &h_star, const ScalarNumber tau) const
        -> state_type
    {
      state_type result;
 
      const auto g = gravity();
      const auto n = manning_friction_coefficient();
 
      const auto h_inverse = inverse_water_depth_mollified(U);
 
      const auto m = momentum(U);
      const auto v_norm = (m * h_inverse).norm();
      const auto factor = ScalarNumber(2.) * g * n * n * v_norm;
 
      const auto denominator = h_star + std::max(h_star, tau * factor);
      const auto denominator_inverse = ScalarNumber(1.) / denominator;
 
      for (unsigned int d = 0; d < dim; ++d)
        result[d + 1] = -factor * denominator_inverse * m[d];
 
      return result;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::nodal_source(
        const PrecomputedVector &pv,
        const unsigned int i,
        const state_type &U_i,
        const ScalarNumber tau) const -> state_type
    {
      const auto &[eta_m, h_star] =
          pv.template get_tensor<Number, precomputed_type>(i);
 
      return manning_friction(U_i, h_star, tau);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::nodal_source(
        const PrecomputedVector &pv,
        const unsigned int *js,
        const state_type &U_j,
        const ScalarNumber tau) const -> state_type
    {
      const auto &[eta_m, h_star] =
          pv.template get_tensor<Number, precomputed_type>(js);
 
      return manning_friction(U_j, h_star, tau);
    }
 
 
    template <int dim, typename Number>
    template <typename ST>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::expand_state(const ST &state) const
        -> state_type
    {
      using T = typename ST::value_type;
      static_assert(std::is_same_v<Number, T>, "template mismatch");
 
      constexpr auto dim2 = ST::dimension - 1;
      static_assert(dim >= dim2,
                    "the space dimension of the argument state must not be "
                    "larger than the one of the target state");
 
      state_type result;
      result[0] = state[0];
      for (unsigned int i = 1; i < dim2 + 1; ++i)
        result[i] = state[i];
 
      return result;
    }
 
    template <int dim, typename Number>
    template <typename ST>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::from_initial_state(
        const ST &initial_state) const -> state_type
    {
      const auto primitive_state = expand_state(initial_state);
      return from_primitive_state(primitive_state);
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::from_primitive_state(
        const state_type &primitive_state) const -> state_type
    {
      const auto &h = primitive_state[0];
 
      auto state = primitive_state;
      /* Fix up momentum: */
      for (unsigned int i = 1; i < dim + 1; ++i)
        state[i] *= h;
 
      return state;
    }
 
 
    template <int dim, typename Number>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::to_primitive_state(
        const state_type &state) const -> state_type
    {
      const auto h_inverse = inverse_water_depth_sharp(state);
 
      auto primitive_state = state;
      /* Fix up velocity: */
      for (unsigned int i = 1; i < dim + 1; ++i)
        primitive_state[i] *= h_inverse;
 
      return primitive_state;
    }
 
 
    template <int dim, typename Number>
    template <typename Lambda>
    DEAL_II_ALWAYS_INLINE inline auto
    HyperbolicSystemView<dim, Number>::apply_galilei_transform(
        const state_type &state, const Lambda &lambda) const -> state_type
    {
      auto result = state;
      auto M = lambda(momentum(state));
      for (unsigned int d = 0; d < dim; ++d)
        result[1 + d] = M[d];
      return result;
    }
 
  } // namespace ShallowWater
} // namespace ryujin